Stochastic Dominance Notes AGEC 662

Transcription

1 McCarl July 1996 Stochastic Dominance Notes AGEC 66 A undamental concern, when looking at risky situations is choosing among risky alternatives. Stochastic dominance has been developed to identiy conditions under which one risky outcome would be preerable to another. The basic approach o stochastic dominance is to resolve risky choices while making the weakest possible assumptions. Generally, stochastic dominance assumes an individual is an expected utility maximizer and then adds urther assumptions relative to preerence or wealth and risk aversion. We will discuss stochastic dominance in two parts. First, we will review the basic theory then we will cover a number o the extensions that had been done. 1.1 Background - Assumptions 1.0 Background to Stochastic Dominance There are a number o important assumptions in traditional stochastic dominance. Assumption #1 - individuals are expected utility maximizers. Assumption # - two alternatives are to be compared and these are mutually exclusive, i.e., one or the other must be chosen not a convex combination o both. Assumption #3 - the stochastic dominance analysis is developed based on population probability distributions. 1. Background - The Expected Utility Basis o Stochastic Dominance 1

2 Stochastic dominance assumes expected utility o wealth maximization. Assume x is the level o wealth while (x) and g(x) gives the probability o each level o wealth or alternatives and g. We may then write the dierence in the expected utility between the prospects as ollows. u(x) (x) dx u(x) g(x) dx and this equation can be rewritten as: u(x) ( (x) g(x) ) dx (1) I is preerred to g then the sign o the above equation would be positive. Conversely, i g is preerred to, the sign o the above equation be negative. 1.3 Background - Integration by Parts One o the classical calculus techniques or integration is called integration by parts. The basic integration by parts ormula is: a db ab b da where a and b are unctions o x..0 Basic Stochastic Dominance.1 First Degree Stochastic Dominance Following the developments in Quirk and Soposnik or Fishburn as reviewed in Anderson, we may apply the integration by parts ormula to the last version o the expected utility equation (1). Let us do this by deining an a and b terms which it the integration by parts structure. Namely, let us choose a to be u(x) and b as the dierence between the cumulative density unctions as ollows:

3 where F(x) G(x) a = u(x) b = (F(x) - G(x)) x (x)dx x g(x)dx in turn the dierential terms are: da u (x) dx db ((x) g(x)) dx Notice that under this substitution that adb encompasses the terms in the expected utility equation. Given this substitution the integration o u(x) ( (x) g(x) ) dx equals u(x) ( F(x) G(x) ) u (x) ( F(x) G(x) ) dx We can observe a couple o things about this result. First, let us look at the let hand part. Notice that when the F(x) and G(x) terms are evaluated at x levels o minus ininity they are both zero because we are at the ar let hand tail o the probability distribution where the cumulative probabilities equal zero. Thus, the evaluation at minus ininity is zero. Similarly, when x equals plus ininity since these are cumulative probability distributions both will equal one so we have the utility o plus ininity times a term which equals one minus one which is zero. Thus, the let part 3

4 o the expression is zero. Now let us look at the right part which is: u (x) ( F(x) G(x) ) dx Suppose we try to characterize something about the sign o this term. Remember, i the overall sign is positive then dominates g. We will restrict the sign by adding assumptions. First, suppose that we assume nonsatiation i.e., that more is preerred to less or u'(x) > 0 or all x. Thus, the u'(x) term does not have anything to do with the overall sign o this term as it will always be a positive multiplier. This means this term takes it s sign rom the F(x) - G(x) term. That term gives the dierence between the two cumulative probability distributions. One can then make a second assumption which is that the dierence between F(x) and G(x) is negative or zero or all x. This means that the cumulative probability o distribution o must always lie on or to the right o the cumulative probability distribution o g (Figure 1). Notice in Figure 1 that or a value o x equal to 7 that there is no meaningul area under the (x) distribution but there is under the g(x) distribution. Note, or the point x that there is an area under both distributions but that the area underneath the g distribution (i.e., the area between the line and the horizontal axis integrated rom the beginning o the probability distribution up to the point x) is greater or the g distribution than it is or the distribution. Note, when this is true or all x points and thereore we can conclude that dominates g. What this then does is leads us to the irst degree stochastic dominance rule which is as ollows: Given two probability distributions and g, distribution dominates distribution g by irst degree stochastic dominance when the decision maker has positive marginal utility o wealth or all x 4

5 (u (x)>0) and or all x the cumulative probability under the distribution is less than or equal to the cumulative probability under the g distribution with strict inequality or some x. This requires that or all x the cumulative probability distribution or is always to the right o the cumulative probability distribution or g or that or every x the cumulative probability o getting that level o wealth or higher is greater under than under g. Note, the strict inequality requirement means the distribution cannot be the same. This is not a revolutionary requirement. Some properties are that the mean o is greater than the mean o g and that or every level o probability you make at least as much money under as you do under g. This is clearly a very weak requirement, but allows one to characterize the choices between two risky distributions or every utility maximizer that preers more wealth to less. This is about as weak an assumption as one can make and still resolve some sort o a choice.. Second Degree Stochastic Dominance The above stochastic dominance development while theoretically elegant is not terribly useul. What this means is when one is comparing two crop varieties. What one has to observe is that one crop variety always has to consistently perorm the other. This may not be the case. The next development in stochastic dominance due to Fishburn; Hanoch and Levy; Hadar and Russell; and Hammond involves making an assumption about risk aversion. We do this by again applying integration by parts and setting the ollowing: a u (x) db (F(x) G(x) ) dx so that: 5

6 da u (x) dx b = (F (x) - G (x)) where the terms F and G are the second integral o F and G with respect to x, i.e.: F (x) x x (x)dx x F (x)dx Under these circumstances i we plug in our integration by parts ormula we get the equation. u (x) ( F (x) G (x) ) u (x) ( F (x) G (x) ) dx The ormula above has two parts. Let us address the right hand part o it irst. This contains the second derivative o the utility unction multiplied times the dierence in the integrals o the cumulative probability distributions with a positive sign in ront o it. In order or us to guarantee that dominates g the sign o this whole term must be positive. Second degree stochastic dominance makes two assumptions that render this term positive. First, assume that the second derivative o the utility unction with respect to x is negative everywhere (u (x) < 0). Also, assume that F (x) is less than or equal to G (x) or all x with strict inequality or some x. Under these circumstances we have a negative times a negative leading to a positive. We must also sign the let hand part o the above term. First, add the assumption on nonsatiation u (x) >0. This term then multiplies by F(x) - G (x) which we know is at plus ininity non-positive since we have already assumed F(x) is smaller than G (x) while it is zero at x equals minus ininity since there is no area at that stage. This coupled with the leading minus sign 6

7 means the whole term will be positive. The second degree stochastic dominance rule can now be stated. Under the assumptions that an individual has 1) positive marginal utility; (u (x) >0) ) diminishing marginal utility o income (u (x) >0) and 3) that or all x F (x) is less than or equal to G (x) with strict inequality or some x then we can say that dominates g by a second degree stochastic dominance. One aspect o the above assumptions worth mentioning is that when u (x) is less than zero and u (x) is greater than zero, this implies the Pratt risk aversion coeicient is positive. Also, the area assumption that the integral under the cumulative probability distribution o must be smaller than the integral under g allows the cumulative distributions to cross as long as the dierence in the areas beore they cross is greater than the dierence in their areas ater they cross. Figure shows the case where second degree stochastic dominance would exist. Notice the area between g and beore x equals 11 exceeds that ater x equals 11. Figure 3 shows a case where stochastic dominance cannot be concluded because o the crossing below x equals 9..3 The Extension to the Third Degree Stochastic Dominance Whitmore and Hammond made up a third degree stochastic dominance rule by extending this approach once more. They again apply integration by parts. There they ind i one assumes that the irst derivative is positive, the second derivative negative and the third derivative positive and that the third integral o the probability unction o is always smaller than that o g then dominates g. The logical intuition is that we have a risk averter with diminishing absolute risk aversion. This test has not been used a great deal in the literature. 7

8 .4 Geometric Interpretation Suppose we interpret the irst and second degree tests geometrically. First degree stochastic dominance requires that the cumulative probability distribution o always lies to the right or just touching the probability distribution o g. What then happens is the cumulative probability at each income level under g is greater than or equal to the cumulative probability o reaching that income level or less under the distribution. Conversely one minus that cumulative probability (which is the probability o that income exceeds that level) has to be greater under the distribution than the g distribution. When the distributions cross irst degree dominance is not possible. Thus, at some income levels there is greater probability o exceeding that income level with the g distribution than the. What second degree does is assume risk aversion and allows the marginal utility o income at the lower levels o wealth exceed to overcome the utility o the additional income increments at the higher levels. What we care about then is the cumulative area between and g remain positive everywhere or that when alls below g that it has an advantage and retains that advantage starting rom low x values..5 Empirical Implementations Again, as is the argument in the notes on ormation o probability distributions, one does not usually have ull continuous probability distributions. Generally, these distribution come about in a discrete ashion. The above presentation is entirely in terms o integrals. Let us now develop the ways o computing the areas in terms o discrete steps. The ollowing procedure develops the probability distributions and the related integrals. Step 1 - take the wealth or x outcomes or all the probability distributions and array them rom high to low as is inherent in tables

9 Step - write the relative requencies o observations against each o the x levels or each probability distribution. Note, some o these requencies will usually be zero i or example when an x level is observed under distribution g but not observed under distribution. Step 3 - divide the requencies through by the number o observations under each o the items and i there are 10 observations or each probability would be the relative requency times 1/10th. Step 4 - orm the cumulative probability distribution starting at the irst x value by taking zero plus the probability o that x or each distribution. For the second and all later x values take the cumulative probability or the prior x plus the relative requency and accumulate this and at the end both o the cumulative probability distributions should be a one. The algebraic ormulae or the area is: F o = G o = 0 F = F + i i-1 i G = G + g i i-1 i Where the F and G are the cumulative probability at step i and g and the are the event i i i i probabilities. Step 5 - orm the second integral o the probability using the ormulae; F = 0,1 G = 0,1 F = F + F * ( x - x ) i > 1,i,i-1 i i i-1 G = G + G * ( x - x ) i > 1,i,i-1 i i i-1 Where F i and G i are second integrals at Step i. 9

10 An example o this is given in Table 1. Suppose that or distribution we have one observation at one and three at two, our at our, our at ive, two at six, three at seven, two at nine, and one at ten or a total o 0. For g we have two at one, ive at two, one at three, ive at our, and seven at seven. We then orm the distributions as in the Table. Notice or the distribution we have zero probability o observations at three and eight, whereas or distribution g we have zero probabilities at six through ten. We then orm the cumulative probability distribution unction as in the cd columns and put the integral o the cumulatives as in the last two columns in the Table. In this comparison dominates g by irst degree stochastic dominance since every single observation in the cd column or F is less than or equal to that or G with some strict equalities. Example presents a case where second degree stochastic dominance holds. First degree ails since or the case o x = 5 the cd or is greater than the cd or g. But when we integrate the cumulatives then F is always less than or equal that or G with several strict inequalities. Example 3 shows a case where dominance does not hold. Note here that the integrated cumulative probability distribution or is both larger and smaller than that or g. I one looks at this case careully one can also see that one o the problems with stochastic dominance and that is that the whole reason or the ailure o the dominance tests is the low level crossing at x = Moment Based Stochastic Dominance Analysis One way that stochastic dominance analysis can be done is under distributional assumption. There are a number o derivations o second degree (SSD) results in such cases as reviewed in Pope and Ziemer; Ali; Bawa and Bury. Namely, i one assumes normality then the SSD rule is 10

11 u u g g with at least one strict in inequality where u, u, and are the mean and variance parameters g g o the and g data that is assumed to be normally distributed. Similarly, under log normal distributions we get the rule u u g g g and under Gamma distributions we get the rule 1 max (1, / 1 ) Each o these rules is discussed in Pope and Ziemer 3.0 Problems With Stochastic Dominance While stochastic dominance as presented above seems to have nice properties; it has problems inherent in its assumptions and it is not a very discriminating instrument. Let us shed some light on the diiculties and on approaches and procedures that have been advanced to get around them. 3.1 Non-Discrimination - Low Crossings The irst problem is the lack o ability to discriminate among cases with low crossings. Stochastic dominance requires the dominant distribution to always have a greater minimum than the dominated distribution. I the distribution shows a vast improvement under all the observations but the lowest one as in Figure 3 or Table 3, then stochastic dominance will not hold in any orm. The real question is how risk adverse will individuals be? Stochastic dominance 11

12 assumes that the individuals all in the class o all risk averters which includes ininitely risk adverse individuals. It assumes someone can possess a risk aversion parameter that is so large that the utility o the small dierence at the lowest observation is extraordinary important. The extension to get around this is involves placing bounds on the risk aversion parameter and saying it has to all in particular numerical ranges. 3. Portolio Eects A second assumption o stochastic dominance is the assumption that the alternatives are mutually exclusive. When one does stochastic dominance one ignores the possibility that the alternatives could be diversiied. This is perectly reasonable when one is talking about dealing with two mutually exclusive alternatives. On the other hand, i one is looking at acreages o crops to grow an obvious possibility is to not have a monoculture area but rather have a diversiied area where one can grow some combination o both. One can use stochastic dominance to look at such questions but one has to orm a larger set o mutually exclusive alternatives. For example, 100% corn, 95% corn - 5% cotton, 90% corn - 10% cotton, etc. 3.3 Sample Size A third problem with stochastic dominance is sampling distributions. Namely, when one goes out and inds data, one does not ind population data and one usually inds a sample. For example, data rom 5 years o crop yield experiments whereas the true crop could be exposed to a million dierent years o weather. Stochastic dominance is subject to sampling error and one could or example draw a particular good or bad year or even have some contamination in sample collection. 1

13 4.0 Problem Resolution Each o these problems has had some degree o attention toward its relaxation. We will cover those now. 4.1 Crossings and Dominance Failures Low crossings is a problem in stochastic dominance so is the existence o crossings in general which cause second degree stochastic dominance rule ailure. There have been solutions proposed which make additional assumptions relative to the risk aversion parameters. Two techniques will be reviewed that all into this class Generalized Stochastic Dominance An extension o stochastic dominance that has been utilized is generalized stochastic dominance (GSD). Here one again starts rom the expected utility unction: u (x) ( F(x) G(x) ) dx Meyer investigated the magnitude o this expression under the conditions that the Pratt risk aversion coeicient all into an interval: r 1 (x) u (x) u (x) r (x) In this ramework what we do is look at the utility dierence between and g but we hold the risk aversion parameter in a particular interval. Meyer poses an optimal control ormat or this examination 13

14 Max u (x) ( F(x) G(x) ) dx (u (x)) r 1 (x) u (x) u (x) u (x) u (x) u (x) r (x) () In this problem Meyer chooses u(x) so as to maximize the utility dierence while requiring the risk aversion parameter to be in a particular interval. When this problem is solved i the solution has a negative objective unction value, then under any utility unction choice within the r(x) interval, the expected utility criteria will always be positive and thereore must dominate g. What this says is that when the decision makers utility unction has r(x) is in the interval between r(x) and r (x) that this dominance holds. 1 Meyer recognized that this is a simple optimal control problem since it is linear in the control variables. The problem has what it is called a Bang-Bang solution. Namely, the solution or r(x) is at either r (x) or r (x) depending on the criteria. The criteria developed is as ollows: 1 r(x ) r 1 (x ) i r (x ) i x x u (x) ( F(x) G(x) ) dx > 0 u (x) ( (F(x) G(x) ) dx 0 Which leads to an recursive calculation o the optimal objective unction. X n 1 Q n X n u (x) ( F(x) G(x) ) dx Q n 1 The generalized stochastic dominance rule can now be developed. Namely, dominates g 14

15 whenever the solution o the maxim and in () is positive as calculated by the recursive relationship explained above. This is a numerical test based on the data and means that when one goes through numerical evaluation equation or given and g probability distributions with upper and lower bounds on the risk aversion parameter, that stochastic dominance holds whenever the numerical value o the objective unctions comes out positive. Meyer originally wrote a computer program to do this and McCarl has a related program called MEYEROOT on the class web page. This has been a airly heavily used technique in risk analysis and virtually everyone that has used it has used constants or r (x) and r (x) saying that the risk aversion parameter lies 1 somewhere between two absolute numbers. Note this does not imply that the risk aversion parameter is constant but rather that it could be increasing, decreasing or o any other orm as long as it remains in between the two bounds. The biggest problem in using that technique has always been to ind the r, r values. There is a computer program available on McCarl s web 1 page which does do the Meyer calculations. It can use ixed r, r or will search or the largest 1 interval given a value or r or r. Namely, when given r it searches or the largest r that permits 1 1 dominance or when given r inds the smallest r that still permits dominance. However, there are 1 diiculties in how big r and r should be. There is an alternative approach which can be used as 1 discussed next.. Finally, note GSD is a generalization o the other stochastic dominance orms when r = 0 1 and r = we get test equivalent to second degree while r = - and r = is the same as irst 1 degree. 4. Finding the Discriminating Risk Aversion Parameter - Low Crossings 15

16 Yet another approach has been used to deal with crossings. Hammond showed that given two alternatives which cross once that under constant absolute risk aversion there is a break-even risk aversion coeicient (BRAC) that dierentiates between those two alternatives. Further, anyone with a risk aversion coeicient (RAC) larger than that particular BRAC will preer one alternative while any one with a RAC smaller than the BRAC would preer the other alternative. Hammond s approach has been implemented in two dierent ways. First and undamentally, Hammond noted the expected utility problem given an RAC (r) e rx (x) dx is a orm o the mathematical statistics moment generating unction, i.e., see Hogg and Craig. Moment generating unctions have been derived and are tabled or alternative distributional orms. Perhaps this should be illustrated with an example. Suppose we assume normality and use the moment generating unction or the normal distribution. In this case, the moment generating unction given the risk aversion parameter r or distribution is as ollows: m(r) e (r u r ) I we go to solve this or the break-even risk aversion parameter, irst thing we would do is set the expected utilities equal: e (r u r ) e (r u g g r ). In turn i we take the logs o both sides 16

17 r u r r u g g r this can be manipulated to r ( which yields two roots r = 0 r = g ) (u u g )r o (u u g ) 0 g Notice then or any two normally distributed prospects we can ind a break-even risk aversion parameter using this ormula just using data on the means and variances. Thereore, what one can do is take the moment generating unction or the and g distributions then solve or the BRAC which leads to the expected utility being equal. Subsequently, one would investigate the value o the utility dierence unction above and below that BRAC to come up with a conclusion about which distribution is preerred above and below it. The important dierence in this technique relative to Meyers generalized stochastic dominance is rather than having to speciy a risk aversion parameter bound one, can solve or the BRAC then proceed to investigate whether it is reasonable or individuals to have risk aversion coeicients which are larger or smaller than that particular value. However, this introduces the problem o knowing the unctional orm o the assumed probability distribution. McCarl wrote a program to implement Hammond s approach with an empirical discrete 17

18 distribution o unknown orm. This program is called RISKROOT and is available on the web page. RISKROOT takes data or two alternatives and searches or the break-even risk aversion parameters between those two alternatives. This is done by solving the ollowing equation or all applicable values o r. i e rx i ((x i ) g(x i )) 0 There are several characteristics that are recognized in RISKROOT. First, Hammond shows the number o roots that can be ound is determined by the number o distribution crossings. I there are no distribution crossings then either irst degree stochastic dominance must exist and no BRAC can be ound. I they cross, then one or more BRAC s may be ound. Second, the number o roots depend upon the number o crossings and i there are 5 crossings it is conceivable there will be 5 BRAC s. Intuitively, a case with multiple BRAC s occurs with distributions with a lower minimum and higher maximum than another but where the other distribution has a higher mean. Note, at extremely high risk aversion, the distribution with a higher minimum will be preerred, while at extremely high risk taking the distribution with the higher maximum will be preerred. However, at moderate levels o risk aversion (somewhere around zero) the distribution with the higher mean will be preerred. One would start preerring the distribution with the higher minimum then switch to the distribution with the higher mean then switch back to the distribution with the higher maximum. Thus, there would be two crossings and one would expect to ind two roots. Third, the maximum size o the BRAC examined by RISKROOT is dependent upon a 18

19 ormula derived rom McCarl and Bessler. It is possible that due to very low or high crossings, a risk aversion parameter cannot be ound to be the low or high enough in order to dierentiate among the prospects. Fourth, the BRAC arises rom the solution o i e rx i ((x i ) g(x i )) 0 Note that when the risk aversion parameter equals zero the above unction becomes zero (since the ormula reduces to the sum o the minus g probabilities equaling one minus one) while when -rx r equals ininity the above unction is zero. Thus, (since e goes to zero) there is always a root or risk aversion parameter equal zero and positive ininity. Fith, when one uses RISKROOT to ind BRAC s one inds results which are limiting results on the Meyer GSD. One cannot span a BRAC within Meyer s ramework. Namely, i one ound a BRAC o.1 where distribution is preerred to g above it whereas below it g is preerred to then i one spans.1 unanimous dominance cannot be ound Technique Choice Both the Meyer and GSD technique and the McCarl RISKROOT techniquecan be used to resolve stochastic dominance choices. We recommend that the McCarl RISKROOT technique be used because it identiies the BRAC points at which preerence switches. Let us briely present an argument rom McCarl (1990). At any point or any RAC value one can always ind a Meyer interval which will give the same results as the BRAC preerences. Namely, i a BRAC is ound at 0.1 above which is preerred to g and below which the converse is true. Now suppose one wants to investigate what happens at 0.09 is preerred to then one can ind an interval 19

20 surrounding 0.09 (it may need to be a small one) where according to the Meyer GSD g is preerred to. There is no way with GSD r below 0.1 and 0. anywhere above.1 that one can 1 ever ind GSD results where is preerred to g. Thus, the BRACs deine the places where the preerence shits. The RISKROOT BRAC gives much stronger results than GSD telling exactly where the preerence shits rather than having to make one hunt or appropriate levels o risk aversion bounds to put into the GSD program. 5.0 Sampling Pope and Ziemer investigated sampling error. Not a lot can be said beyond the ollowing 1) when distribution means and variances get close together that the probability o improper dominance conclusions can become quite high. ) using the moment based stochastic dominance rules is inerior to using the empirical distribution based stochastic dominance rules. 3) the smaller the sample size the more likely one is to have errors. 6.0 Portolios A problem in stochastic dominance involves potential presence o a portolios. Namely, one may be looking at two stochastic prospects which are not mutually exclusive but which may be correlated, i.e., i one was looking at two crops one might ind that wheat and corn perorm dierently in dierent weather conditions because they utilize dierent growing seasons. Here we investigate the question o what happens with the correlation. The undamental basis or these notes is in the paper by McCarl, et al. where the portolio problem is investigated. In that paper several results occur which will not be reviewed here where previous authors have shown conditions under which diversiication between two alternatives is optimal. The question that we 0

21 deal with involves the conditions under which when one inds that dominates g that the prospect will also dominate all combinations o the and g. The procedure or investigation that is used in that paper is based on two moment based stochastic dominance rules. This portolio based investigation requires us to take on some additional assumptions so that we may generate analytical results. We will rely on the moment based normality GSD rule which states that normally distributed prospect dominates normally distributed prospect g whenever the ollowing two conditions are discovered. u u g g Now we wish to see when prospect dominates prospect g via the above stochastic dominance rules that we will also ind that prospect will dominate prospect h which is a convex combination o and g. A convex combination is written according to the ollowing ormula: h = + (1 - )g where varies between 0 and 1. We also know rom mathematical statistics that when we orm prospect h that its mean and variance are given by the u h u (1 ) u g h (1 ) g p (1 ) g we also know since dominates g that the ollowing two equations are satisied. u u g g 1

22 Now what we need to do is investigate the more general dominance conditions between and h and try and ind conditions under which those conditions will hold given some arbitrary and g. The irst rule that we will investigate is the relationship between the means. Notice that the deinition o the mean as expressed above allows us to write the ollowing: u h u (1 ) u g u (1 )u u or u h u This arises since u is less than or equal to u. Thus, uniormly u is always less than or equal to u g h so the irst o the two dominance rules is always satisied. Examining the second dominance rule is more complicated. Here we need to investigate the relationship between the variance o and the variance o h. We can get the variance o h rom h (1 ) g ( ) (1 ) g Suppose we make a substitution namely since the variance o is smaller than the variance o g, we can write. which renders our equation into the orm K = g K 1 h (1 ) K ( ) (1 ) K Factoring out the we get the ollowing or now suppose we get = n or

23 ( ( (1 ) K K (1 )) (1 ) K (1 ) K 1 I we collect the square terms in this and use the classical quadratic ormula, we ind the roots are = 1 and (K 1) (K K 1) I we wish to preclude convex combinations we wish the equality o the variances to hold somewhere outside the realm o easible convex combinations. So what we wish to do is that be strictly greater than or equal to one 1. This implies K 1 K K 1 which can be simpliied to and inally to K 1/K = / 1 where is the correlation coeicient. Thus, we have the restriction that the correlation coeicient must be greater than or equal to the ratio o the variances. The signiicance o this result is that we now have a condition under which we are certain that i dominates g via a second degree stochastic dominance then will dominate all potential convex combinations o and g. This equation has several other implications. Namely, i the items are perectly correlated 3

24 then we are always sae because we know that is always less than or equal to. Thus, i 1 = 1 it is always going to be greater than the ratio o the standard errors. Similarly, i is zero or negative then there is no way that one can ever guarantee that all the convex combinations are dominated. McCarl, et al. do rather extensive evaluation on this rule in mind that it works in a very high proportion i the cases or normal and non-normal cases. They also develop a second criteria or dominance. This starts rom a rule the dierences in mean. This values use o the certain equivalent or the normal distribution stating dominates g whenever u r r u h h under this rule ollowing the same approach as talked through above, they ind the ollowing condition will dominate all combinations o and g whenever g (u u g ) r g This can be transormed using the rule that r is twice the value as explained in McCarl and Bessler as ollows: r Z to become g (u u g ) Z g 4

25 what this rule shows is that the maximum acceptable correlation coeicient becomes smaller as the means become more disparable. What these rules can be used or is to examine when one has two potentially diversed alternatives whether can successully do dominance analysis between the two without considering diversiications. Namely, i the rules are satisied one is sae i the rule is not satisied then one needs to potentially consider diversiications. One can also use the ormula or as expressed above giving a particular ratio o the standard errors and a correlation coeicient to ind the largest possible diversiication that should be considered. For example, i one plugs in the ratio o Z = or where g is twice as big as, with a correlation o.5 then one can use the ormula to ind that the diversiication that should be considered is something between 100% o and 43% o and one then can lay a grid out where one might consider 100, 90, 80, 70, 60, 50 and 43% o and corresponding values o g and then do stochastic dominance over all those alternatives. 5

26 Reerences Anderson, J.R Risk eiciency in the interpretation o agricultural production research. Rev. Mktg. Agric. Econ. 4(3): Fishburn, P.C Decision and Value Theory. New York: Wiley. Hadar, J., and W.R. Russell Rules or ordering uncertain prospects. Am. Econ. Rev. 59(1):5-34. Hanoch, G., and H. Levy The eiciency analysis o choice involving risk. Rev. Econ. Stud. 36(3): Meyer, Jack. Choice Among Distributions. Journal o Economic Theory. 14(1977): McCarl, B.A. Generalized Stochastic Dominance: An Empirical Examination. Southern Journal o Agricultural Economics.,(December 1990): McCarl, B.A. Preerence Among Risky Prospects Under Constant Risk Aversion. Southern Journal o Agricultural Economics. 0,(December 1988):5-33. McCarl, B.A., Thomas O. Knight, James R. Wilson, and James B. Hastie. Stochastic Dominance Over Potential Portolios: Caution Regarding Covariance. American Journal o Agricultural Economics. 69,4(November 1987): Pope, Rulon D. and Rod F. Ziemer. Stochastic Eiciency, Normality, and Sampling Errors in Agricultural Risk Analysis. American Journal o Agricultural Economics. 66,1(February 1984): Quirk, J.P., and R. Saposnik Admissibility and measurable utility unctions. Rev. Econ. Stud. 9(): Whitmore, G.A Third-degree stochastic dominance. Am. Econ. Rev. 60(3):

27 Table 1. First Degree Stochastic Dominance Example x Freq Freq g Pd Pd g CDF CDF g Intcd Intcd g () i (g) i (F) i (G) i (F i) (G i) Mean Std Err

28 Table. Second Degree Stochastic Dominance Example x Freq Freq g Pd Pd g CDF CDF g Intcd Intcd g () (g) (F) (G) (F ) (G ) i i i i i i Mean Std Err.4.9 8

29 Table 3. No Stochastic Dominance x Freq Freq g Pd Pd g CDF CDF g Intcd Intcd g () (g) (F) (G) (F ) (G ) i i i i i i % 0% 5% 0% % 35% 0% 35% % 5% 0% 40% % 10% 40% 50% % 35% 60% 85% % 0% 70% 85% % 15% 85% 100% % 0% 85% 100% % 0% 95% 100% % 0% 100% 100% Mean Std Err

30 Table 4. Example OUTPUT FROM RISKROOT - CONSTANT RISK AVERSION ROOT FINDER Example 1 DISTRIBUTION 1 NAME IS CASE 1 DISTRIBUTION NAME IS CASE THE DISTRIBUTIONS DO NOT CROSS -- 1 IS DOMINANT Example THE DISTRIBUTION CDFS CROSS TIMES 1 HAS BEEN FOUND DOMINANT BETWEEN HAS BEEN FOUND DOMINANT BETWEEN Example 3 SUMMARY STATISTICS ON THE DATA DISTRIBUTION MEAN STDDEV MIN MAX CASE CASE RAC IS LIMITED TO BE BETWEEN +/ E+01 BASED ON MCCARL AND BESSLER THE DISTRIBUTION CDFS CROSS 3 TIMES 1 HAS BEEN FOUND DOMINANT BETWEEN TROUBLE -- FOUND 1 DOMINANT AT HIGHEST RAC -- SHOULD FIND RAC LARGE ENOUGH THAT DOMINATED 1 HAS BEEN FOUND DOMINANT BETWEEN

31 31

32 Figure 1. First Degree 1. Cumulative Probability Distribution Distribution g Wealth 3

33 1 Figure. Second Degree 1. Cumulative Probability Distribution Distribution g Wealth 33